Problem: The hyperbola \[-x^2+2y^2-10x-16y+1=0\]has two foci. Find the coordinates of either of them. (Enter your answer as an ordered pair. Enter only one of the foci, not both.)
Solution: To find the standard form for the equation of the hyperbola, we complete the square in both variables: \[\begin{aligned} -(x^2+10x) + 2(y^2-8y) + 1 &= 0 \\ -(x^2+10x+25) + 2(y^2-8y+16) + 1 &= -25 + 32 \\ -(x+5)^2 + 2(y-4)^2 &= 6 \\ \frac{(y-4)^2}{3} - \frac{(x+5)^2}{6} &= 1. \end{aligned}\]This fits the standard form of the hyperbola \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\]where $a=\sqrt{3},$ $b=\sqrt{6},$ $h=-5,$ and $k=4.$ Thus, the center of the hyperbola is the point $(h,k)=(-5, 4).$ Because the $y^2$ coefficient is positive and the $x^2$ coefficient is negative, the foci are vertically aligned with the center of the hyperbola. We have \[c = \sqrt{a^2 + b^2} = \sqrt{3+6} = 3,\]which is the distance from the center of the hyperbola to each focus. Therefore, the two foci of the hyperbola are $(-5, 4 \pm 3),$ which gives two points: $\boxed{(-5, 7)}$ and $\boxed{(-5, 1)}.$ (Either point is an acceptable answer.)[asy]
void axes(real x0, real x1, real y0, real y1)
{
	draw((x0,0)--(x1,0),EndArrow);
    draw((0,y0)--(0,y1),EndArrow);
    label("$x$",(x1,0),E);
    label("$y$",(0,y1),N);
    for (int i=floor(x0)+1; i<x1; ++i)
    	draw((i,.1)--(i,-.1));
    for (int i=floor(y0)+1; i<y1; ++i)
    	draw((.1,i)--(-.1,i));
}
path[] yh(real a, real b, real h, real k, real x0, real x1, bool upper=true, bool lower=true, pen color=black)
{
	real f(real x) { return k + a / b * sqrt(b^2 + (x-h)^2); }
    real g(real x) { return k - a / b * sqrt(b^2 + (x-h)^2); }
    if (upper) { draw(graph(f, x0, x1),color,  Arrows); }
    if (lower) { draw(graph(g, x0, x1),color,  Arrows); }
    path [] arr = {graph(f, x0, x1), graph(g, x0, x1)};
    return arr;
}
void xh(real a, real b, real h, real k, real y0, real y1, bool right=true, bool left=true, pen color=black)
{
	path [] arr = yh(a, b, k, h, y0, y1, false, false);
    if (right) draw(reflect((0,0),(1,1))*arr[0],color,  Arrows);
    if (left) draw(reflect((0,0),(1,1))*arr[1],color,  Arrows);
}
void e(real a, real b, real h, real k)
{
	draw(shift((h,k))*scale(a,b)*unitcircle);
}
size(8cm);
axes(-12,3,-2, 10);
yh(sqrt(3),sqrt(6),-5,4,-11.8,1.8);
dot((-5,4)^^(-5,7)^^(-5,1));
[/asy]